AMC 8 · 2009 · #7

Grade 6 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates

coordinate-geometryarea-triangles coordinate-geometryidentify-subproblems ↑ Prerequisites: coordinate-geometryarea-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A triangular plot of land has corners A, C, and D. A lies 3 miles west of point B on Main Street (east-west). C lies 3 miles north of B on the railroad (north-south). D lies another 3 miles north of C on the railroad. Find the area of triangle ACD in square miles.

Pick an answer.

(A)

(B)

(C)

4.5

(D)

(E)

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Two perpendicular roads make this a coordinate-plane problem in disguise. Tool #1 (Draw a Diagram) says: put B at the origin, Main Street on the x-axis, and the railroad on the y-axis. Then A, C, D all have clean integer coordinates. With C and D both on the y-axis, side CD is vertical, so the perpendicular distance from A to that side is just AB — Tool #7 (Identify Subproblems) splits the area question into two easy subproblems: "how long is CD?" and "how far is A from the railroad?" Tool #16 (Count the Complement) gives a check: △ ACD = △ ABD - △ ABC.

1STEP 1

Put B at the origin, Main Street on the x-axis and the railroad on the y-axis, giving A = (-3, 0), C = (0, 3), and D = (0, 6).

A = (-3, 0), C = (0, 3), D = (0, 6)

2STEP 2

Take CD as the base: C and D both sit on the y-axis, so CD is vertical with length 6 - 3 = 3 miles.

CD = 6 - 3 = 3 miles

3STEP 3

The height from A to line CD (the y-axis) is A's distance from that line, |-3|, so h = 3 miles — exactly segment AB.

h = |-3| = 3 miles

4STEP 4

Apply $\frac{1}{2}$ · base · height with base 3 and height 3: area = 4.5 square miles → (C).

Area =

\frac{1}{2}

· base · height =

\frac{1}{2}

· 3 · 3 = 4.5 sq mi → (C)

Answer

4.5

Sanity-check the size. Triangle ABD (corners at (-3,0), (0,0), (0,6)) is a right triangle with legs 3 and 6, so its area is

\frac{1}{2}

· 3 · 6 = 9 sq mi — the largest answer choice (E). The plot ACD chops off the bottom half (△ ABC has legs 3 and 3, area 4.5), so △ ACD should be the leftover 9 - 4.5 = 4.5 sq mi. That matches (C), and it sits comfortably between 3 and 6 — exactly where we'd expect the upper half of the big triangle to fall.

💡Key takeaway

This AMC 8 problem only needs Grade 6 coordinate-plane area: pick a side as the base, measure the perpendicular distance from the opposite vertex, and use $\frac{1}{2}$ · base · height.

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